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<title>Simulations for Statistical and Thermal Physics</title>

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<h3 style="text-align:center;">Monte Carlo simulation of hard disks</h3>

<p class="header_title">Introduction</p>



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<p>The application of the Metropolis algorithm for a system of hard disks can be stated very simply:</p>

<p>1. Choose a particle at random and generate trial a change in its x and y coordinates:</p>
<p class="center">
x(i) = x(i) + (2r - 1)&#948;</p>
<p class="center">
y(i) = y(i) + (2r - 1)&#948;,
</p>
<p>where r is a uniform random number in the unit interval and &#948; is the maximum displacement.</p>

<p>2. Accept the trial move if the trial position of the disk does not overlap another disk. Otherwise, the move is rejected and
the old configuration is retained. A reasonable, although not necessarily optimum choice for &#948; is to choose its value
such that approximately 20% of the trial moves are accepted.</p>

<p>The program uses units such that the diameter &#963; = 1.</p>

<p class="header_title">Problems</p>

<p>1. The main quantity of interest is the radial distribution function g(r). Describe its qualitative r-dependence. How does g(r) change with increasing density?</p>

<p>2. How does the form of g(r) compare to that a system of particles interacting with the lennard-Jones potential at the same density?</p>

<p>3. The pressure P of a system of hard disks is related to the value of g(r) at contact by the expression</p>
<p class="center">
<img src="pressurehd.jpg" alt="" align="middle" >.
</p>
<p>Because the hard disks rarely touch, it is difficult to obtain good statistics for g(r) at contact. Fit the values of g(r) close to r = &#963; to a second-order polynomial in r and extrapolate the values of g(r) for r greater than r = &#963;<sup>+</sup>.</p>

<p>4. What does the phase diagram of a system of hard disks look like? Does the system become a solid at high densities? Start the system at a low density and slowly compress the system. We do so by first determining the minimum distance between the centers of any two disks. Because this distance cannot be less than &#963;, its value bounds the amount that we can compress the system. We multiply this distance by the parameter &#955; = <tt>scale lengths</tt> after every Monte Carlo step per particle. Start the system in a rectangular configuration and then compress the system by setting <tt>scale lengths = 0.95</tt>. Is the system a fluid or a solid at high densities? If its a solid, what is its symmetry?</p>

<p class="header_title">Java Classes</p>

<ul>

<li>HD</li>
<li>HDMCApp</li>

</ul>

<p class = "small">Updated 13 May 2008.</p>

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